For now I converted the paths to the Linux format, but still need to make this multiplatform in the .pro file.
The project still does not build: http://pastebin.com/raw/KmT1NnhN
Tons of errors outside of the project source code. This builds just fine under MingW. Is it incompatible with GCC 4.8.3?

$ gcc --version
gcc (GCC) 4.8.3 20140911 (Red Hat 4.8.3-7)
Copyright (C) 2013 Free Software Foundation, Inc.
This is free software; see the source for copying conditions. There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Notice the path C:/Program\ Files\ (x86)/AMD\ APP\ SDK/3.0/include.
First this Windows OS specific path shouldn't be there when you compile the project on Unix based system.
So now that it is included accidentally when the compiler encounters this path, it is unable to parse it due to space or special characters in the path.
In case of Qt this user defined path is actually added from projects .pro file.
So you have to modify the pro file so that the platform dependent paths are picked up accordingly or just remove these Windows OS based paths for quick test.